# machine learning – Probability Density Function

A probability distribution P over X × {0, 1}. P can be defined in term of its marginal distribution over X , which we will denote by $$P_X$$ and the conditional labeling distribution, which is defined by the regression function
$$µ(x) = P_{ (x,y)∼P} (y = 1 | x)$$
consider a 2-dimensional Euclidean domain, that is $$X = R^2$$, and the
following process of data generation: The marginal distribution over X is uniform over two square areas (1, 2) × (1, 2) ∪ (3, 4) × (1.5, 2.5). Points in the first
square Q1 = (1, 2) × (1, 2) are labeled 0 (blue) and points in the second square
Q2 = (3, 4) × (1.5, 2.5) are labeled 1 (red)

Describe the density function of $$P_X$$, and the regression function, Bayes predictor and Bayes risk of P.

In the image, I have defined the Probability density function. Is that correct? I want to find it over both the axis and in general in two dimensions.